# 20777_Problem_solving_Year_6_Paterns_and_algebra_Fractions_and_decimals_Using_units_of_measurement

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YEAR 6<br />

PROBLEM-SOLVING<br />

IN MATHEMATICS<br />

Patterns <strong>and</strong> <strong>algebra</strong>/<br />

<strong>Fractions</strong> <strong>and</strong> <strong>decimals</strong>/<br />

<strong>Using</strong> <strong>units</strong> <strong>of</strong> <strong>measurement</strong><br />

Two-time winner <strong>of</strong> the Australian<br />

Primary Publisher <strong>of</strong> the <strong>Year</strong> Award

<strong>Problem</strong>-<strong>solving</strong> in mathematics<br />

(Book G)<br />

Published by R.I.C. Publications ® 2008<br />

Copyright © George Booker <strong>and</strong><br />

Denise Bond 2007<br />

RIC–<strong>20777</strong><br />

This master may only be reproduced by the<br />

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Website: www.ricpublications.com.au<br />

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FOREWORD<br />

Books A–G <strong>of</strong> <strong>Problem</strong>-<strong>solving</strong> in mathematics have been developed to provide a rich resource for teachers<br />

<strong>of</strong> students from the early years to the end <strong>of</strong> middle school <strong>and</strong> into secondary school. The series <strong>of</strong> problems,<br />

discussions <strong>of</strong> ways to underst<strong>and</strong> what is being asked <strong>and</strong> means <strong>of</strong> obtaining solutions have been built up to<br />

improve the problem-<strong>solving</strong> performance <strong>and</strong> persistence <strong>of</strong> all students. It is a fundamental belief <strong>of</strong> the authors<br />

that it is critical that students <strong>and</strong> teachers engage with a few complex problems over an extended period rather than<br />

spend a short time on many straightforward ‘problems’ or exercises. In particular, it is essential to allow students<br />

time to review <strong>and</strong> discuss what is required in the problem-<strong>solving</strong> process before moving to another <strong>and</strong> different<br />

problem. This book includes extensive ideas for extending problems <strong>and</strong> solution strategies to assist teachers in<br />

implementing this vital aspect <strong>of</strong> mathematics in their classrooms. Also, the problems have been constructed <strong>and</strong><br />

selected over many years’ experience with students at all levels <strong>of</strong> mathematical talent <strong>and</strong> persistence, as well as<br />

in discussions with teachers in classrooms, pr<strong>of</strong>essional learning <strong>and</strong> university settings.<br />

<strong>Problem</strong>-<strong>solving</strong> does not come easily to most people,<br />

so learners need many experiences engaging with<br />

problems if they are to develop this crucial ability. As<br />

they grapple with problem, meaning <strong>and</strong> find solutions,<br />

students will learn a great deal about mathematics<br />

<strong>and</strong> mathematical reasoning; for instance, how to<br />

organise information to uncover meanings <strong>and</strong> allow<br />

connections among the various facets <strong>of</strong> a problem<br />

to become more apparent, leading to a focus on<br />

organising what needs to be done rather than simply<br />

looking to apply one or more strategies. In turn, this<br />

extended thinking will help students make informed<br />

choices about events that impact on their lives <strong>and</strong> to<br />

interpret <strong>and</strong> respond to the decisions made by others<br />

at school, in everyday life <strong>and</strong> in further study.<br />

Student <strong>and</strong> teacher pages<br />

The student pages present problems chosen with a<br />

particular problem-<strong>solving</strong> focus <strong>and</strong> draw on a range<br />

<strong>of</strong> mathematical underst<strong>and</strong>ings <strong>and</strong> processes.<br />

For each set <strong>of</strong> related problems, teacher notes <strong>and</strong><br />

discussion are provided, as well as indications <strong>of</strong><br />

how particular problems can be examined <strong>and</strong> solved.<br />

Answers to the more straightforward problems <strong>and</strong><br />

detailed solutions to the more complex problems<br />

ensure appropriate explanations, the use <strong>of</strong> the<br />

pages, foster discussion among students <strong>and</strong> suggest<br />

ways in which problems can be extended. Related<br />

problems occur on one or more pages that extend the<br />

problem’s ideas, the solution processes <strong>and</strong> students’<br />

underst<strong>and</strong>ing <strong>of</strong> the range <strong>of</strong> ways to come to terms<br />

with what problems are asking.<br />

At the top <strong>of</strong> each teacher page, there is a statement<br />

that highlights the particular thinking that the<br />

problems will dem<strong>and</strong>, together with an indication<br />

<strong>of</strong> the mathematics that might be needed <strong>and</strong> a list<br />

<strong>of</strong> materials that could be used in seeking a solution.<br />

A particular focus for the page or set <strong>of</strong> three pages<br />

<strong>of</strong> problems then exp<strong>and</strong>s on these aspects. Each<br />

book is organised so that when a problem requires<br />

complicated strategic thinking, two or three problems<br />

occur on one page (supported by a teacher page with<br />

detailed discussion) to encourage students to find<br />

a solution together with a range <strong>of</strong> means that can<br />

be followed. More <strong>of</strong>ten, problems are grouped as a<br />

series <strong>of</strong> three interrelated pages where the level <strong>of</strong><br />

complexity gradually increases, while the associated<br />

teacher page examines one or two <strong>of</strong> the problems in<br />

depth <strong>and</strong> highlights how the other problems might be<br />

solved in a similar manner.<br />

R.I.C. Publications ® www.ricpublications.com.au <strong>Problem</strong>-<strong>solving</strong> in mathematics<br />

iii

FOREWORD<br />

Each teacher page concludes with two further aspects<br />

critical to successful teaching <strong>of</strong> problem-<strong>solving</strong>. A<br />

section on likely difficulties points to reasoning <strong>and</strong><br />

content inadequacies that experience has shown may<br />

well impede students’ success. In this way, teachers<br />

can be on the look out for difficulties <strong>and</strong> be prepared<br />

to guide students past these potential pitfalls. The<br />

final section suggests extensions to the problems to<br />

enable teachers to provide several related experiences<br />

with problems <strong>of</strong> these kinds in order to build a rich<br />

array <strong>of</strong> experiences with particular solution methods;<br />

for example, the numbers, shapes or <strong>measurement</strong>s<br />

in the original problems might change but leave the<br />

means to a solution essentially the same, or the<br />

context may change while the numbers, shapes or<br />

<strong>measurement</strong>s remain the same. Then numbers,<br />

shapes or <strong>measurement</strong>s <strong>and</strong> the context could be<br />

changed to see how the students h<strong>and</strong>le situations<br />

that appear different but are essentially the same<br />

as those already met <strong>and</strong> solved. Other suggestions<br />

ask students to make <strong>and</strong> pose their own problems,<br />

investigate <strong>and</strong> present background to the problems<br />

or topics to the class, or consider solutions at a more<br />

general level (possibly involving verbal descriptions<br />

<strong>and</strong> eventually pictorial or symbolic arguments).<br />

In this way, not only are students’ ways <strong>of</strong> thinking<br />

extended but the problems written on one page are<br />

used to produce several more problems that utilise<br />

the same approach.<br />

Mathematics <strong>and</strong> language<br />

The difficulty <strong>of</strong> the mathematics gradually increases<br />

over the series, largely in line with what is taught<br />

at the various year levels, although problem-<strong>solving</strong><br />

both challenges at the point <strong>of</strong> the mathematics<br />

that is being learned as well as provides insights<br />

<strong>and</strong> motivation for what might be learned next. For<br />

example, the computation required gradually builds<br />

from additive thinking, using addition <strong>and</strong> subtraction<br />

separately <strong>and</strong> together, to multiplicative thinking,<br />

where multiplication <strong>and</strong> division are connected<br />

conceptions. More complex interactions <strong>of</strong> these<br />

operations build up over the series as the operations<br />

are used to both come to terms with problems’<br />

meanings <strong>and</strong> to achieve solutions. Similarly, twodimensional<br />

geometry is used at first but extended<br />

to more complex uses over the range <strong>of</strong> problems,<br />

then joined by interaction with three-dimensional<br />

ideas. Measurement, including chance <strong>and</strong> data, also<br />

extends over the series from length to perimeter, <strong>and</strong><br />

from area to surface area <strong>and</strong> volume, drawing on<br />

the relationships among these concepts to organise<br />

solutions as well as giving an underst<strong>and</strong>ing <strong>of</strong> the<br />

metric system. Time concepts range from interpreting<br />

timetables using 12-hour <strong>and</strong> 24-hour clocks while<br />

investigations related to mass rely on both the concept<br />

itself <strong>and</strong> practical <strong>measurement</strong>s.<br />

The language in which the problems are expressed is<br />

relatively straightforward, although this too increases<br />

in complexity <strong>and</strong> length <strong>of</strong> expression across the books<br />

in terms <strong>of</strong> both the context in which the problems<br />

are set <strong>and</strong> the mathematical content that is required.<br />

It will always be a challenge for some students<br />

to ‘unpack’ the meaning from a worded problem,<br />

particularly as problems’ context, information <strong>and</strong><br />

meanings exp<strong>and</strong>. This ability is fundamental to the<br />

nature <strong>of</strong> mathematical problem-<strong>solving</strong> <strong>and</strong> needs to<br />

be built up with time <strong>and</strong> experiences rather than be<br />

iv<br />

<strong>Problem</strong>-<strong>solving</strong> in mathematics www.ricpublications.com.au R.I.C. Publications ®

FOREWORD<br />

diminished or left out <strong>of</strong> the problems’ situations. One<br />

reason for the suggestion that students work in groups<br />

is to allow them to share <strong>and</strong> assist each other with<br />

the tasks <strong>of</strong> discerning meanings <strong>and</strong> ways to tackle<br />

the ideas in complex problems through discussion,<br />

rather than simply leaping into the first ideas that<br />

come to mind (leaving the full extent <strong>of</strong> the problem<br />

unrealised).<br />

An approach to <strong>solving</strong> problems<br />

Try<br />

an approach<br />

Explore<br />

means to a solution<br />

Analyse<br />

the problem<br />

The careful, gradual development <strong>of</strong> an ability to<br />

analyse problems for meaning, organising information<br />

to make it meaningful <strong>and</strong> to make the connections<br />

among them more meaningful in order to suggest<br />

a way forward to a solution is fundamental to the<br />

approach taken with this series, from the first book<br />

to the last. At first, materials are used explicitly to<br />

aid these meanings <strong>and</strong> connections; however, in<br />

time they give way to diagrams, tables <strong>and</strong> symbols<br />

as underst<strong>and</strong>ing <strong>and</strong> experience <strong>of</strong> <strong>solving</strong> complex,<br />

engaging problems increases. As the problem forms<br />

exp<strong>and</strong>, the range <strong>of</strong> methods to solve problems<br />

is carefully extended, not only to allow students to<br />

successfully solve the many types <strong>of</strong> problems, but<br />

also to give them a repertoire <strong>of</strong> solution processes<br />

that they can consider <strong>and</strong> draw on when new<br />

situations are encountered. In turn, this allows them<br />

to explore one or other <strong>of</strong> these approaches to see<br />

whether each might furnish a likely result. In this way,<br />

when they try a particular method to solve a new<br />

problem, experience <strong>and</strong> analysis <strong>of</strong> the particular<br />

situation assists them to develop a full solution.<br />

Not only is this model for the problem-<strong>solving</strong> process<br />

helpful in <strong>solving</strong> problems, it also provides a basis for<br />

students to discuss their progress <strong>and</strong> solutions <strong>and</strong><br />

determine whether or not they have fully answered<br />

a question. At the same time, it guides teacher<br />

questions <strong>of</strong> students <strong>and</strong> provides a means <strong>of</strong> seeing<br />

underlying mathematical difficulties <strong>and</strong> ways in<br />

which problems can be adapted to suit particular<br />

needs <strong>and</strong> extensions. Above all, it provides a common<br />

framework for discussions between a teacher <strong>and</strong><br />

group or whole class to focus on the problem-<strong>solving</strong><br />

process rather than simply on the solution <strong>of</strong> particular<br />

problems. Indeed, as Alan Schoenfeld, in Steen L (Ed)<br />

Mathematics <strong>and</strong> democracy (2001), states so well, in<br />

problem-<strong>solving</strong>:<br />

getting the answer is only the beginning rather than<br />

the end … an ability to communicate thinking is<br />

equally important.<br />

We wish all teachers <strong>and</strong> students who use these<br />

books success in fostering engagement with problem<strong>solving</strong><br />

<strong>and</strong> building a greater capacity to come to<br />

terms with <strong>and</strong> solve mathematical problems at all<br />

levels.<br />

George Booker <strong>and</strong> Denise Bond<br />

R.I.C. Publications ® www.ricpublications.com.au <strong>Problem</strong>-<strong>solving</strong> in mathematics<br />

v

CONTENTS<br />

Foreword .................................................................. iii – v<br />

Contents .......................................................................... vi<br />

Introduction ........................................................... vii – xix<br />

A note on calculator use ................................................ xx<br />

Teacher notes................................................................... 2<br />

Surface area...................................................................... 3<br />

Volume <strong>and</strong> surface area................................................. 4<br />

Surface area <strong>and</strong> volume................................................. 5<br />

Teacher notes................................................................... 6<br />

The farmers market.......................................................... 7<br />

Teacher notes................................................................... 8<br />

Pr<strong>of</strong>it <strong>and</strong> loss.................................................................. 9<br />

Calculator patterns........................................................ 10<br />

Puzzle scrolls.................................................................. 11<br />

Teacher notes................................................................. 12<br />

Weather or not............................................................... 13<br />

Showtime....................................................................... 14<br />

Probably true.................................................................. 15<br />

Teacher notes................................................................. 16<br />

Wilderness explorer....................................................... 17<br />

Teacher notes................................................................. 18<br />

Changing lockers............................................................ 19<br />

Cycle days...................................................................... 20<br />

Puzzle scrolls.................................................................. 21<br />

Teacher notes................................................................. 22<br />

Office hours.................................................................... 23<br />

At the <strong>of</strong>fice................................................................... 24<br />

Out <strong>of</strong> <strong>of</strong>fice................................................................... 25<br />

Teacher notes................................................................. 26<br />

Number patterns............................................................ 27<br />

Teacher notes................................................................. 28<br />

Magic squares............................................................... 29<br />

Sudoku........................................................................... 30<br />

Alphametic puzzles........................................................ 31<br />

Teacher notes................................................................. 32<br />

At the shops................................................................... 33<br />

The plant nursery........................................................... 34<br />

On the farm.................................................................... 35<br />

Teacher notes................................................................. 36<br />

Fisherman’s wharf.......................................................... 37<br />

Teacher notes................................................................. 38<br />

Making designs.............................................................. 39<br />

Squares <strong>and</strong> rectangles................................................. 40<br />

Designer squares........................................................... 41<br />

Teacher notes................................................................. 42<br />

How many?..................................................................... 43<br />

How far?......................................................................... 44<br />

How much?..................................................................... 45<br />

Teacher notes................................................................. 46<br />

Prospect Plains............................................................... 47<br />

Teacher notes................................................................. 48<br />

Money matters............................................................... 49<br />

Scoring points................................................................ 50<br />

Puzzle scrolls.................................................................. 51<br />

Teacher notes.................................................................. 52<br />

Riding to work................................................................. 53<br />

Bike tracks....................................................................... 54<br />

Puzzle scrolls................................................................... 55<br />

Teacher notes.................................................................. 56<br />

Farm work....................................................................... 57<br />

Farm produce.................................................................. 58<br />

Selling fish...................................................................... 59<br />

Teacher notes.................................................................. 60<br />

Rolling along................................................................... 61<br />

Solutions .................................................................62–71<br />

Isometric resource page .............................................. 72<br />

0–99 board resource page ........................................... 73<br />

4-digit number exp<strong>and</strong>er resource page (x 3) .............. 74<br />

10 mm x 10 mm grid resource page ........................... 75<br />

15 mm x 15 mm grid resource page ............................ 76<br />

Triangular grid resource page ...................................... 77<br />

vi<br />

<strong>Problem</strong>-<strong>solving</strong> in mathematics www.ricpublications.com.au R.I.C. Publications ®

TEACHER NOTES<br />

<strong>Problem</strong>-<strong>solving</strong><br />

To use strategic thinking to solve problems<br />

Materials<br />

grid paper, counters in several different colours<br />

Focus<br />

These pages explore more complex problems in which the<br />

most difficult step is to find a way <strong>of</strong> coming to terms with<br />

what the problem is asking. <strong>Using</strong> a table or diagram to<br />

explore the situation will assist in seeing all the conditions<br />

that need to be considered.<br />

Discussion<br />

Page 19<br />

The first problem can be solved using counters on a<br />

grid or colouring the squares to see what is happening.<br />

1 2 3 4 5 6 7 8 9<br />

The number <strong>of</strong> possibilities can then be seen directly or<br />

patterns can be sought.<br />

Analysis <strong>of</strong> the patterns shows that the squares represent<br />

the factors <strong>of</strong> the number <strong>of</strong> the column in which they<br />

occur. Determining the factors <strong>of</strong> each number 1–50 shows<br />

that 48 has the most factors or entries in a column. The<br />

other questions are solved by considering prime numbers,<br />

squares <strong>of</strong> prime numbers <strong>and</strong> systematically examining<br />

the pairs <strong>of</strong> factors in each number. A discussion <strong>of</strong> prime<br />

numbers is on page 64.<br />

Page 20<br />

For the first problem, consider the distances covered each<br />

15 minutes (a table or list would help). After 15 minutes,<br />

Jane would walk 1 km <strong>and</strong> have 15 km left, while Jenny<br />

would ride 3 km <strong>and</strong> have 13 km left. After 30 minutes,<br />

Jane would walk 2 km <strong>and</strong> have 14 km left, while Jenny<br />

would ride 6 km <strong>and</strong> have 10 km left. After 45 minutes,<br />

Jane would walk 3 km <strong>and</strong> have 13 km left, while Jenny<br />

would ride 9 km <strong>and</strong> have only 7 km left. Jenny would<br />

then be home first. Jenny should leave the bicycle after 30<br />

minutes <strong>and</strong> they would both arrive home together after<br />

150 minutes. (Tables to illustrate this solution are on page<br />

64).<br />

For the second problem, organising the information on to a<br />

table will supply a solution. The slower speed requires an<br />

additional 10 km, the faster speed covers 15 km too much:<br />

hours<br />

distance @<br />

10 km/h<br />

distance<br />

to park<br />

distance @<br />

15 km/h<br />

distance<br />

to park<br />

1 10 20 15 0<br />

2 20 30 30 15<br />

3 30 40 45 30<br />

4 40 50 60 45<br />

5 50 60 75 60<br />

After 5 hours, the distance is 60 km so 12 km/h would<br />

arrive at the exact time. The third problem is done the<br />

same way – the required speed is 9.6 km/h for 5 hours.<br />

Page 21<br />

The puzzle scrolls contain a number <strong>of</strong> different problems<br />

all involving strategic thinking to find possible solutions. In<br />

most cases students will find tables, lists <strong>and</strong> diagrams are<br />

needed to manage the data while exploring the different<br />

possibilities. For example, the investigation about bread<br />

rolls could involve students list the multiples <strong>of</strong> 60 in a<br />

table with the total to make $11 <strong>and</strong> seeing which is a<br />

multiple <strong>of</strong> 85.<br />

Multiple<br />

<strong>of</strong> $0.60<br />

Possible difficulties<br />

• Not using a diagram or table to come to terms with<br />

the problem conditions<br />

• Unable to see how to connect the time cycled to the<br />

distance travelled<br />

• Considering only some aspects <strong>of</strong> the puzzle scrolls<br />

Extension<br />

• Change the numbers <strong>and</strong> the scenarios to write other<br />

problems based on the puzzle scrolls.<br />

• Use different speeds <strong>and</strong> times for the problems<br />

on p. 20.<br />

Look for a<br />

multiple <strong>of</strong> $0.85<br />

Total<br />

$.060 $10.40 $11.00<br />

$1.20 $9.80 $11.00<br />

18<br />

<strong>Problem</strong>-<strong>solving</strong> in mathematics www.ricpublications.com.au R.I.C. Publications ®

CHANGING LOCKERS<br />

The local college has exactly 1000 students, each <strong>of</strong> whom has a locker. The lockers<br />

are along the passageways <strong>and</strong> numbered 1–1000. To raise money for local charities,<br />

the students organised a competition with an entry fee <strong>of</strong> $1.00.<br />

All 1000 students had to run past, opening or shutting locker doors:<br />

• the first student opened the door <strong>of</strong> every locker<br />

• the second student closed every locker door with an even number<br />

• the third student changed every third locker, closing those that were open <strong>and</strong><br />

opening those that were closed<br />

• the fourth student changed every fourth locker, <strong>and</strong> so on.<br />

1. The student or students who could predict ahead <strong>of</strong> time which lockers would be open<br />

would nominate the charity that would receive the $1000.00.<br />

(a) What would you predict?<br />

(e) Use your pattern to work out which<br />

lockers would be open after all<br />

1000 students had run past.<br />

(b) Would any lockers remain open<br />

after 10 students had passed along<br />

the rows <strong>of</strong> lockers?<br />

(f) Can you find some lockers that<br />

only changed twice?<br />

(c) Which lockers would be open<br />

after 50 students had changed 50<br />

lockers?<br />

(g) Can you see a pattern for the<br />

numbers on these lockers?<br />

(d) Can you see a pattern for the<br />

numbers <strong>of</strong> the open lockers?<br />

(h) What is the largest number on a<br />

locker that changed only twice?<br />

R.I.C. Publications ® www.ricpublications.com.au <strong>Problem</strong>-<strong>solving</strong> in mathematics<br />

19

CYCLE DAYS<br />

Jane <strong>and</strong> Jenny rode their bicycles to the bay for a picnic. On the way back, when they<br />

were 16 km from home, the front wheel on Jane’s bicycle was damaged in a large<br />

pothole <strong>and</strong> could no longer be ridden. In order for them both to get home in good<br />

time, they decided to share the walking <strong>and</strong> bike riding.<br />

1. At first, Jane would walk while Jenny would ride her bicycle. After some time, Jenny<br />

would leave her bicycle at the side <strong>of</strong> the road <strong>and</strong> continue on foot. When Jane<br />

reached the bicycle, she would then ride it home. Jane walks at 4 km per hour <strong>and</strong><br />

cycles at 10 km per hour, while Jenny walks at 5 km per hour <strong>and</strong> cycles at 12 km per<br />

hour. How long should Jenny ride her bicycle for both <strong>of</strong> them to arrive home at the<br />

same time?<br />

2. When Jane’s bicycle was repaired, they decided to go for another bike ride. This time,<br />

they would set <strong>of</strong>f separately <strong>and</strong> meet at the Jetty at midday, then set up their picnic<br />

in the park next door. Jenny knew that if she took it easy <strong>and</strong> rode at 10 km per hour,<br />

she would be an hour late, but if she really pushed herself <strong>and</strong> rode at 15 km per hour,<br />

she would arrive an hour early. At what speed should she ride in order to arrive just on<br />

time?<br />

3. Jane did not have quite as far to travel so she knew she did not have to ride as hard as<br />

Jenny. If she rode at 12 km per hour, she would arrive an hour early, but if she rode at<br />

8 km per hour, she would be an hour late. At what speed should Jane ride to get there<br />

at the same time as Jenny?<br />

4. At what time did each girl leave her home?<br />

20<br />

PUZZLE SCROLLS<br />

1. $65 000 is divided among 5<br />

brothers with each one getting<br />

$3000 more than their younger<br />

sibling. How much will the<br />

youngest brother get?<br />

2. A rectangular table is twice as<br />

long as it is wide. If it was 3 m<br />

shorter <strong>and</strong> 3 m wider it would<br />

be a square. What size is the<br />

table?<br />

3. How many blocks would you<br />

need to make this stack <strong>of</strong> cubes<br />

into a cube 6 blocks high?<br />

4. I spent $11.00 at the bakery<br />

buying bread rolls for 60c <strong>and</strong><br />

85c. How many <strong>of</strong> each roll did I<br />

buy?<br />

5. How <strong>of</strong>ten in a 12-hour period<br />

does the sum <strong>of</strong> the digits on a<br />

digital clock equal 9?<br />

6. A triangle has a perimeter <strong>of</strong> 80<br />

cm. Two <strong>of</strong> its sides are equal<br />

<strong>and</strong> the third side is 8 cm more<br />

than the equal sides.<br />

03:24<br />

ON<br />

OFF<br />

GIGA-BLASTER<br />

What is the length <strong>of</strong> the third<br />

side?<br />

21

SOLUTIONS<br />

Note: Many solutions are written statements rather than just numbers. This is to encourage teachers <strong>and</strong> students to solve<br />

problems in this way.<br />

CHANGING LOCKERS................................................ page 19<br />

1. (a) Answers will vary.<br />

Colour squares or place counters on a grid to show<br />

when a door is open:<br />

A door is changed an odd number <strong>of</strong> times to<br />

remain open. These locker numbers are square<br />

number (one number as a factor twice, so an odd<br />

number <strong>of</strong> factors): 1, 4, 9, 16, 25, 49, ...<br />

A prime number has only two factors <strong>and</strong> these<br />

lockers will always be closed.<br />

(b) 1, 4, 9<br />

(c) 1, 4, 9, 16, 25, 49<br />

(d) Square numbers have an odd number <strong>of</strong> factors <strong>and</strong><br />

these lockers will be open.<br />

(e) 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169,<br />

225, 256, 289, 324, 361, 400, 441, 484, 529, 576,<br />

625, 676, 729, 784, 841, 900, 961<br />

(f) 2, 3, 5, 7, 11, 13<br />

(g)<br />

(h)<br />

1 2 3 4 5 6 7 8 9<br />

They are prime numbers <strong>and</strong> have only 2 factors<br />

The Sieve <strong>of</strong> Eratosthenes shows that after 2 <strong>and</strong><br />

3, all primes must be 1 more or less than a multiple<br />

<strong>of</strong> 6 (although some <strong>of</strong> these numbers are not<br />

prime – this was discussed in Book F).<br />

1 2 3 4 5 6 7<br />

8 9 10 11 12 13<br />

14 15 16 17 18 19<br />

20 21 22 23 24 25<br />

26 27 28 29 30 31<br />

32 33 34<br />

=166 x 6 = 996 is the closest multiple <strong>of</strong> 6 to 1000,<br />

so 997 is the largest prime.<br />

997 is the largest locker number that is only<br />

changed twice.<br />

CYCLE DAYS............................................................... page 20<br />

1. Looking at their rates <strong>of</strong> walking <strong>and</strong> cycling, in 15<br />

minutes Jane walks 1 km <strong>and</strong> Jenny cycles 3 km:<br />

Jane walking<br />

minutes<br />

Jenny cycling<br />

distance<br />

walked<br />

distance left to<br />

cycle<br />

They can arrive home together after 2 hours 30 minutes.<br />

Jenny cycles for 30 minutes <strong>and</strong> walks the remaining 10<br />

km in 2 hours. Jane walks for 90 minutes to reach the<br />

bicycle <strong>and</strong> cycles the remaining 10 km in 1 hour.<br />

Jenny should ride her bicycle for 30 minutes<br />

time taken<br />

15 1 15 1 hour 45 minutes<br />

30 2 14 1 hour 54 minutes<br />

45 3 13 2 hours 3 minutes<br />

60 4 12 2 hours 12 minutes<br />

75 5 11 2 hours 21 minutes<br />

90 6 10 2 hours 30 minutes<br />

minutes<br />

distance<br />

cycled<br />

distance left to<br />

walk<br />

time taken<br />

15 3 13 2 hours 51 minutes<br />

30 6 10 2 hours 30 minutes<br />

45 9 7 2 hours 90 minutes<br />

60 12 4 1 hour 48 minutes<br />

Jane 90 minutes walking 1 hour cycling<br />

km 0 1 2 3 4 5 6 7 8 9 10 11 12 13 15 15 16<br />

Jenny 30 minutes cycling 2 hours walking<br />

2. The slower speed requires an additional 10 km, the<br />

faster speed covers 15 km too much:<br />

hours<br />

distance @<br />

10km/her<br />

distance to<br />

park<br />

distance @<br />

15 km/her<br />

distance<br />

to park<br />

1 10 20 15 0<br />

2 20 30 30 15<br />

3 30 40 45 30<br />

4 40 50 60 45<br />

5 50 60 75 60<br />

After 5 hours, the distance to the park at each speed is<br />

the same – 60 km.<br />

She would arrive at the exact time cycling at<br />

12 km per hour.<br />

64<br />

SOLUTIONS<br />

Note: Many solutions are written statements rather than just numbers. This is to encourage teachers <strong>and</strong> students to solve<br />

problems in this way.<br />

3. The slower speed requires an additional 8 km, the faster<br />

speed cover 12 km too much:<br />

hours<br />

distance @<br />

10km/her<br />

distance to<br />

park<br />

distance @<br />

15 km/her<br />

distance<br />

to park<br />

1 10 20 15 0<br />

2 20 30 30 15<br />

3 30 40 45 30<br />

4 40 50 60 45<br />

5 50 60 75 60<br />

After 5 hours, the distance to the park at each speed is<br />

the same – 48 km.<br />

She would arrive at the exact time cycling at 9.6 km per<br />

hour<br />

4. Both Jenny <strong>and</strong> Jane should leave at 7:00 am<br />

PUZZLE SCROLLS ...................................................... page 21<br />

1. $7000<br />

2. width 6 m, length 12 m<br />

3. 210 blocks<br />

4. 7 @ 60 c, 8 @ 85 c<br />

5. 57 times. Student answers will vary <strong>and</strong> it is unlikely<br />

they will find them all.<br />

6. 32 cm<br />

65

10 mm x 10 mm GRID RESOURCE PAGE<br />

75

15 mm x 15 mm GRID RESOURCE PAGE<br />

76<br />